Write in Standard Form 25x^2-39y^2+150x+390y=-225

$25 x^{2} - 39 y^{2} + 150 x + 390 y = - 225$

Complete the square for $25 x^{2} + 150 x$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 25$

$b = 150$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{150}{2 \cdot 25}$

Simplify the right side.

Cancel the common factor of $150$ and $2$ .

Factor $2$ out of $150$ .

$d = \frac{2 \cdot 75}{2 \cdot 25}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 25$ .

$d = \frac{2 \cdot 75}{2 (25)}$

Cancel the common factor.

$d = \frac{2 \cdot 75}{2 \cdot 25}$

Rewrite the expression.

$d = \frac{75}{25}$

Cancel the common factor of $75$ and $25$ .

Factor $25$ out of $75$ .

$d = \frac{25 \cdot 3}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$d = \frac{25 \cdot 3}{25 (1)}$

Cancel the common factor.

$d = \frac{25 \cdot 3}{25 \cdot 1}$

Rewrite the expression.

$d = \frac{3}{1}$

Divide $3$ by $1$ .

$d = 3$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{150^{2}}{4 \cdot 25}$

Simplify the right side.

Simplify each term.

Raise $150$ to the power of $2$ .

$e = 0 - \frac{22500}{4 \cdot 25}$

Multiply $4$ by $25$ .

$e = 0 - \frac{22500}{100}$

Divide $22500$ by $100$ .

$e = 0 - 1 \cdot 225$

Multiply $- 1$ by $225$ .

$e = 0 - 225$

Subtract $225$ from $0$ .

$e = - 225$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(x + 3)}^{2} - 225$ .

$25 {(x + 3)}^{2} - 225$

Substitute $25 {(x + 3)}^{2} - 225$ for $25 x^{2} + 150 x$ in the equation $25 x^{2} - 39 y^{2} + 150 x + 390 y = - 225$ .

$25 {(x + 3)}^{2} - 225 - 39 y^{2} + 390 y = - 225$

Move $- 225$ to the right side of the equation by adding $225$ to both sides.

$25 {(x + 3)}^{2} - 39 y^{2} + 390 y = - 225 + 225$

Complete the square for $- 39 y^{2} + 390 y$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 39$

$b = 390$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{390}{2 \cdot - 39}$

Simplify the right side.

Cancel the common factor of $390$ and $2$ .

Factor $2$ out of $390$ .

$d = \frac{2 \cdot 195}{2 \cdot - 39}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 39$ .

$d = \frac{2 \cdot 195}{2 (- 39)}$

Cancel the common factor.

$d = \frac{2 \cdot 195}{2 \cdot - 39}$

Rewrite the expression.

$d = \frac{195}{- 39}$

Cancel the common factor of $195$ and $- 39$ .

Factor $39$ out of $195$ .

$d = \frac{39 (5)}{- 39}$

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$d = - 5$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{390^{2}}{4 \cdot - 39}$

Simplify the right side.

Simplify each term.

Raise $390$ to the power of $2$ .

$e = 0 - \frac{152100}{4 \cdot - 39}$

Multiply $4$ by $- 39$ .

$e = 0 - \frac{152100}{- 156}$

Divide $152100$ by $- 156$ .

$e = 0 - - 975$

Multiply $- 1$ by $- 975$ .

$e = 0 + 975$

Add $0$ and $975$ .

$e = 975$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 39 {(y - 5)}^{2} + 975$ .

$- 39 {(y - 5)}^{2} + 975$

Substitute $- 39 {(y - 5)}^{2} + 975$ for $- 39 y^{2} + 390 y$ in the equation $25 x^{2} - 39 y^{2} + 150 x + 390 y = - 225$ .

$25 {(x + 3)}^{2} - 39 {(y - 5)}^{2} + 975 = - 225 + 225$

Move $975$ to the right side of the equation by adding $975$ to both sides.

$25 {(x + 3)}^{2} - 39 {(y - 5)}^{2} = - 225 + 225 - 975$

Simplify $- 225 + 225 - 975$ .

Add $- 225$ and $225$ .

$25 {(x + 3)}^{2} - 39 {(y - 5)}^{2} = 0 - 975$

Subtract $975$ from $0$ .

$25 {(x + 3)}^{2} - 39 {(y - 5)}^{2} = - 975$

Flip the sign on each term of the equation so the term on the right side is positive.

$- 25 {(x + 3)}^{2} + 39 {(y - 5)}^{2} = 975$

Divide each term by $975$ to make the right side equal to one.

$- \frac{25 {(x + 3)}^{2}}{975} + \frac{39 {(y - 5)}^{2}}{975} = \frac{975}{975}$

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{{(y - 5)}^{2}}{25} - \frac{{(x + 3)}^{2}}{39} = 1$

Write in Standard Form 25x^2-39y^2+150x+390y=-225

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